Dimension-independent simplification and refinement of Morse complexes

Ascending and descending Morse complexes, determined by a scalar field f defined over a manifold M, induce a subdivision of M into regions associated with critical points of f, and compactly represent the topology of M. We define two simplification operators on Morse complexes, which work in arbitrary dimensions, and we define their inverse refinement operators. We describe how simplification and refinement operators affect Morse complexes on M, and we show that these operators form a complete set of atomic operators to create and update Morse complexes on M. Thus, any operator that modifies Morse complexes on M can be expressed as a suitable sequence of the atomic simplification and refinement operators we have defined. The simplification and refinement operators also provide a suitable basis for the construction of a multi-resolution representation of Morse complexes.

► We define simplification operators on Morse complexes in arbitrary dimensions. ► Simplification operators reduce the incidence relation on Morse complexes, and reduce the number of cells in Morse–Smale complexes at each step. ► We define the inverse refinement operators. ► Simplification and refinement operators form a complete set of basis operators for modifying Morse complexes. ► Simplification and refinement operators form a basis for building a multi-resolution representation of Morse complexes.